How do I fix this face shadow glitch?
Here is how to do it:
Your current mesh is having a pretty bad topology since you cut those circles into it.

First of all navigate to Edit &gt; User Preferences &gt; Addons and enable Mesh:Loop Tools

Select a couple of faces in a square shape (where you want the circles to be) and click the RMB &gt; Loop tools &gt; circle.


Voila!! You have perfect circles with a good topology...


BUT...Here too you will see those sharp edges. But unlike yours it can be smoothed out by giving a subdivision-surface modifier of level 2.
You are now free to extrude it to make the tentacles.